193
1 INTRODUCTION
Theultimatestrengthofanintactshiphasproviding
an important approach to ensure the structure will
not collapse under maximum expected bending
moment, torsional loading or combined action of
loads. For an ageing thin walled structure, it is
vulnerable to various types of defects and damages
induced by different phenomena such as corrosion
and fat
igue cracking. In addition to ship’s intact
strength,itisnecessarytohaveanassessmentofthe
residual ultimate strength of ship structures in
damaged conditions. Cracks of any size may be
produced at various locations and orientations
throughoutplatesintheprocessofshipoperationdue
to corrosion and fat
igue damage which will lead to
thereductionofultimateloadcarryingcapacityofthe
hullstructure.Givingexactpredictionoftheresidual
ultimate strength has great significance to avoid
catastrophic failures of damaged structures over the
lifetimeandtoavoiduneconomicaloverdesign.
In recent years, the residual ul
timate strength of
stiffened or unstiffened panels with cracks under
compression, tension, edge shear loading or torsion
have been widely studied using numerical and
experimental methods and related prediction
formulas have been proposed(Hu 2004, Alinia
2007a,b,Alinia2008,Paik2008,Paik2009,Rahmanet
al. 2011, Rahm
an et al. 2013). The required mesh
refinements around the crack and the influence of
relativegeometricalandmechanicalcharacteristicsof
crackedpanelsontheresidualultimatestrengthhave
been investigated and related prediction formulas
havebeenproposed.
Invaluable data can be obtained from physical
experiments for validating theoretical modeling
a
pproaches and demonstrating how structures
behave under closely damaged loading conditions.
Howeverdestructivetestingoflargescalestructures,
such as ships and bridges, are normally limited by
size, complexity of structure and cost constraints.
These factors have placed a great emphasis on
Residual Ultimate Strength of Box Girders with
Variable Cracks
L.Ao&D.Wang
StateKeyLaboratoryofOceanEngineering,ShanghaiJiaoTongUniversity,Shanghai,China
ABSTRACT:Theaimofthepresentstudyistoinvestigatetheresidualultimatestrengthcharacteristicsofbox
girders with variable cracks under torsional loading. A series of finite element models are established by
changingthecracklengthandcrackangleusingacommercia
lFEAprogram,ABAQUS.Thecracksarelocated
atthecenterandtorquesareappliedonbothendsoftheboxbeam.Differentaspectratiosareconsideredto
evaluatetheeffects ofcracksonboxbeamsforvariouswidths andlengthsofpanelsinthemiddle yielding
region.TheaccuracyofthenonlinearFEAresultsisverifiedbyacomparisonwithpreviouspredictedformulas.
BasedontheFEAresults,therelat
ionshipbetweentheresidualultimatestrengthandcrackparameterscanbe
indicatedinafunctionwithperiodofintheformofFourierseries.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 9
Number 2
June 2015
DOI:10.12716/1001.09.02.05
194
developing more effective simplified models and
robust theoretical techniques to examine structural
characteristics.Forships, boxgirders areextensively
usedonhullstructureresearchasthehullismainly
consistedofthinplates,beamsandotheraggregates.
Atpresent,thestudyofboxbeamultimatestrengthis
concentrated inthe
considerationof the influence of
initialimperfectionwhileresearchesconcerningabout
crackingdamagearestillless.
Paik(2001)andSun(2003)investigatedtheultimate
strengthcharacteristicsofshiphullswithlargehatch
openingsundertorsionrespectivelyinnumericaland
experimental ways and the ultimate strength of
cracked open box girders subjected to
variant loads
wasresearchedbyShi(2012)andasimpleprediction
modeloftheresidualultimatestrengthwasproposed.
Kim(2008) investigated the ultimate strength
interaction between bending and torsion of
rectangularsteelboxbeamsconsideringtheeffectsof
residual stresses and initial imperfections applying
different aspect ratios, width‐to‐thickness ratios and
yield stresses by a nonlinear numerical approach.
Simple forms of prediction equations for ultimate
strengths were proposed by a means of regression
analysisonthenumericalresults.
The ultimate strength of cracked box girders
subjected to pure torque was investigated by
Shi(2012)andrelatedsimplemodelforpredictingthe
ultimate
strengthreductionoftheboxgirdersdueto
cracking damage in various crack sizes and crack
locationswasproposed.
Considering the arbitrarily shaped path of crack
propagation for cracked marine structures(SUMI
1998, Okawa 2006), it necessary to involve the
influenceofcrackinclinationonshipstructureswhich
hasnotbeeninvestigated
before.Groupsofboxbeam
models are established through changing crack
parameters such as various crack sizes and inclined
angles and the strength variation regularity of box
beams under different cracking damage forms is
investigated.FEAresultsshowthatcracklengthand
anglehavelargeeffectonthevariationof
theresidual
ultimate strength and the relationship between the
strengthoftheboxgirderandthecrackparametersis
indicatedbysimplifiedpredictionformulasbasedon
thefiniteelementnumericalresults.
2 FINITEELEMENTMODELS
The whole calculation in this paper is conducted
using commercial software ABAQUS 6.11. The
nonlinear shell
finite element S4R is used for
modeling thin plates as a general four nodes shell
elementincludingbothreductionofintegralmethod
and hourglass control mode which can be used for
large deformation analysis of thin plates. The RIKS
algorithm is applied as a method of incremental‐
solution to trace
the proper collapse and post‐
buckling process in structural nonlinear analysis
whichisawidelyusedmethodinstructuralnonlinear
analysis because of advantage of overcoming the
difficulties of traditional Newton method across
critical points during structural nonlinear buckling
equilibrium path and automatically adjusting
incrementalstepsduringiterativeprocesses.
To analyze
the influence of different forms of
cracksonresidualultimatestrengthofboxgirders,A
model with 1/2 +1+1/2 transverse frame spacing as
shown in Figure 1 was used for all non‐linear FE
analysesinthepresentstudy forboxgirders sothat
themiddlepartcanbethe
yieldingregiontoensure
the damage happens in the middle part first and
lateral parts were loading regions. One crack was
consideredtobedistributedinthecenterofeachplate
ofboxbeams(fourcracksintotalwiththesamesize)
and the cracks were presumed to be through
thickness,
havingnofrictionbetweentheiredgesand
no propagation was allowed. The plate initial
deflectionwasnotconsideredforsimplicity.Without
specialstated,the box lengthandwidthwereset as
a=b=1000mm,theboxthicknesswasfixedatt=10mm.
Both material and geometrical nonlinearities were
included in the study. Materials
were considered to
behave in an elasto‐plastic manner having bi‐linear
stress–strainrelationship.Unlessotherwisespecified,
thedefaultmaterial,formostpartsofthework,was
assumed to be mild steel with Young’s modulus,
yieldstressandPoisson’sratiogivenby:
205800EMPa
, 345
y
M
Pa
and 0.3
Inordertofacilitatetherelatedanalysis,theeffects
ofstiffenersorsurroundingmembersontheultimate
strength of box beams were not considered and the
Von Mises yield criterion, known to be the most
suitableoneformildsteel,wasusedthroughoutthis
research.
Figure1.Geometrymodelofboxbeamsundertorque.
The box model was meshed with uniform gird
generally and local mesh refinement was taken
aroundthecracks.Figures 2‐3 showthe variation of
the torsional strength behavior for different mesh
sizes and crack widths considering the possible
influence of mesh size and crack width on
computationalresults.Itis
observedameshsize b
no higher than 20mm is good enough for accurate
results. Different crack width values within certain
limitshavelittleeffectontheultimatestrengthofthe
girderwhichcanbeneglected.Inthiscase,thedefault
gape size between crack faces
l
is taken as
4lmm
for most part of the work considering the
convenience of meshing around the cracks and the
total gird size is taken as
20bmm
and
/1/50bb
whichcangivesatisfiedresults.
The dimension of the transverse frame can have
effects on the ultimate strength of cracked girders
195
possiblybecausetheframeplaysasupportingrolefor
the longitudinal components. Figure 4 shows the
variationofbox’sultimatestrengthastheframesize
rangesfromh=0mmtoh=160mm.Itcanbeseenthat
changing the frame size has little influence on the
ultimate strength when increasing to h=80mm.
A
critical coefficient of frame size can be defined as
/
cc
hhb
andthecriticalvalueofboxframecanbe
givenas
cc
hhb
fordifferentcrosssections.
3 THEULTIMATESTRENGTHOFBOXGIRDERS
UNDERTORSIONBENDING
The residual ultimate strength of a series of cracked
boxbeamsinlargetorsiondeflectionwasinvestigated
by varying the crack length and angle through
nonlinear finite element analysis. As to relate the
length of cracks
to the dimensions of box section
width
b ,thecracklengthwasvariedasthe ratioof
the width
b given by ( l
) 0.2b , 0.3b , 0.4b , 0.5b
whilethecrackinclinationchangedas(
) 0
, 15
,
30
, 45
, 60
, 75
and 90
. The crack was
consideredtobeperpendiculartotheaxisofthebox
whenangletakenas
0
.Theboundarycondition
wasassumedassimplysupportedonbothendsofthe
boxgirder. Inthetorsionalcenter of the ends of the
box beam, one side was constrained as
0ux uy uz
, the other was constrained as
0ux uy,whiledirectionwasalongtheaxisofthe
box. The torque was imposed by giving a torsional
displacement on both ends of the box beam as
0.015rad
, the conversion formula between
bendingmomentandshearstressisasfollowed,
2
T
A
t
(1)
whereTistheTorqueappliedateveryendofthebox
girder,Aistheenclosedareaoftheboxcrosssection,
tistheboxwallthickness.
Figure2. Effect of crack width on the ultimate torsional
strengthofcrackedbox(=45°).
Figure3. Effect of mesh size on the ultimate torsional
strengthofcrackedbox(=45°).
Figure4. Effect of frame size on the ultimate torsional
strengthofcrackedbox.
Figure5showstheaveragetorsionalshearstress‐
twistinganglecurvewhenvaryingtheangleofcrack
in different related crack length
/lb. It is observed
from figure 5 the crack inclination has much more
impact on the boxʹs strength as the crack length
increases.Theincreasementofangleenhancesthebox
strength within certain limits for a certain length.
Changing the angle
has little influence on the
strengthofboxbeamswhilehavingtheleastlengthas
/0.2lb
, where the reduction ratio is only
min
/3.87%
between the min angle and max
90
. On the contrary, the angle has great impact
ontheboxultimatestrengthasthemin‐maxchanging
ratio reaches to
min
/ 35.5%
while large crack
size is given at
/0.5lb
. It is shown from the
diagramthattheultimatetorsionalshearstresshasno
linearproportionalrelationshipwiththeinclinationat
certaincracklengthsasthestresschangeslittlealong
with the angle
changing around 0
and 90
,
howeveritrangesalotwhen
isaround.
Figure 6 shows the distribution of membrane
stresswhenthecrackanglevariesasthecracklength
fixed at
/0.3lb
(Figs 6a，b，c are corresponding
to
0
, 30
, 60
). The left figure shows the
membrane stress component in x direction
196
S11(namely
x
) and the right shows the membrane
stresscomponentinydirectionS22(namely
y
).Itis
observed that the compressive stress mainly focuses
on the crack tips and then develops along the crack
while tensile stress develops from the tip towards
outside. Certain angle is fixed between the box
longitudinal axis and tensile stress in spite of the
variation of crack angles which may
have little
influenceonthestressdistribution.
Figure5.Thetorsionalshearstress‐twistinganglecurveby
changing the crack inclination in varying related crack
lengths.
Duetothepracticalpurposes,thelengthofbeam
panels may vary. Figure 7 shows the influence of
differentaspectratiosontheboxultimatestrengthto
evaluatetheeffectsofcracksonboxbeamsforvarious
widths(a) and lengths(b) of panels in the middle
yielding region. However, the value of
aspect ratio
was taken equal to 1 when other parameters were
concerned. It can be seen that the relation between
/
uY
and /ab will be expressed by a formula of
0.06
0
/(a/b)
uY
.
0u
is the ultimate shear stress
whentheaspectratioistakento1.
4 SIMPLIFIEDPREDICTIONFORMULASFOR
ULTIMATESTRENGTHOFBOXBEAM
The following formula gives the reduction factor of
torsional ultimate strength for box girders with
transversecracks:
00
42
2
1
4
uc
f
u
bct
A
c
R
A
bt b
(2)
where
f
R is the reduction factor of torsional
ultimate strength,
u
is the ultimate torsional shear
stress for box beam with cracks,
0u
is the ultimate
torsional shear stress for perfect girder,
t is the
panelthickness,isthesectionwidthoftheboxgirder,
2c isthecracklength.
Figure6. Membrane stress distribution corresponding for
different crack angles at crack size of l/b=0.3: (a)
0
，
(b)
30
，(c) 60
.
Figure7. Effect of aspect ratio on the ultimate torsional
strengthofcrackedbox(=45°).
Ithasbeenrecognizedthattheproposedformula
in Eq.(2) yields conservative values for the ultimate
torsional strength of box girders with transverse
cracks (
0
)(Shi 2012). Figure 8 compares the
torsional ultimate strength of cracked boxgirdersas
obtained from the FEA and from Eq.(2) by varying
crack lengths and angles where the solid line
representsforthepredictedresultsofEq.(2)andthe
dotted lines for results from the FEM. It shows
conservative results
still can be obtained by Eq.(2)
eventhoughthelengthandinclinationangleofcrack
happentochangeandresultsfromformulainEq.(2)
are parallel to that from numerical calculation by
ABAQUS in general as
0
. However, the
predictedresultshavelargedifferencecomparedwith
theFEAresultsalongwiththeincreasementofcrack
anglewhichcanʹtbereflectedbyequation(2).Results
from Eq.(2) are smaller than numerical results
because the prime formula is derived based on an
197
isolated thin plate while the four panels are not
independentplatesbutfixedtogetherforboxbeams.
Itisclearthattheultimatestrengthandcracklength
haveacertainlinearproportionalrelationshipinspite
ofthevariationofangle.
Considering the geometric symmetry of cracks
with respect to the
box beam, it can be treated the
angle
is larger than
90
for the torque direction
reverse. For example,
30
with the torque
directionreverse can be replaced by
150
where
thedirectiondidnotchange.Takingintoaccountthe
triangular characteristic reflected in the numerical
results,theultimatestrengthreductioncharacteristics
of cracked box beam can be indicated as a function
with period of
in the form of Fourier series as
following:
1.07 0.77 0.29 0.042 sin 2
0.08 0.016 cos 4
f
ll
R
bb
l
b
(3)
where
sin( ) 0.884
, cos( ) 0.386
, /lbis relative
cracklength,
isthecrackinclinationangle.
Figure8. Comparison of ultimate torsional strength
reduction between results from Eq.(2) and that from FEA
forvariouscrackparameters.
Figure 9 shows the relationship between the
torsional reduction factor and crack parameters in a
wholecycle.Itisfoundthatthelargestultimateshear
stress
(max)u
is corresponding to crack angle
75
while the minimum corresponding to
165
with the crack size of
/0.3lb
. The dotted
linesinfigure 9are calculatedresultsobtainedfrom
Eq.(3) which agrees well with the numerical results
fromFEA . However, it should be noticed that both
Eq.(2)andEq.(3) arebasedonthebeammodelwith
cracksofthesamesizeoneachofallfoursides.
The
slenderness ratio of girder plates
(/t) /E
y
b
may have effects on the beam ultimate strength
corresponding to different beam widths(b) which
needtobeconfirmedforfurtherinvestigationandthe
formula validity need to be verified for beams with
cracksonlyfordeckorsideswhicharenotpresented
inthispaper.
Figure9. Ultimate torsional strength reduction
characteristicsinawholecycleoffordifferentcracksizes.
5 CONCLUSION
In this paper, groups of box beam models under
torque force were established by changing the crack
length
l and angle
. The influence of crack
parameters and torsional direction on box ultimate
strength was investigated using nonlinear finite
element method and following conclusions can be
drawn:
1 It is clear that when the crack size is less than
/0.2lb
,changingcrackanglehaslittleinfluence
ontheultimatestrengthofaboxgirderwherethe
reduction ratio is only
min
/3.87%
between
the min angle
0
and max 90
. However,
the angle has great impact on the box ultimate
strength when crack length increases to
/0.5lb
asthemin‐maxchangingratioreachesto
min
/ 35.5%
.
2 Itisobservedthattheultimatestrengthandcrack
sizehaveacertainlinearproportionalrelationship
in spite of the variation of angle. However, no
linearrelationshipbutatrigonometricrelationship
happensbetween the ultimate strengthandcrack
angle especially when relative crack length
/0.2lb .
3 Considering the geometric symmetry of cracks
with respect to the box beam and the triangular
characteristics with changing of crack angle, the
ultimate strength reduction characteristics of box
girdersduetocrackingdamagecanbeindicatedin
afunctionwithperiodof
intheformofFourier
series. It is found that the largest ultimate shear
stress
(max)u
is corresponding to crack angle
75
while the minimum corresponding
to
165
withcracksizeof /0.3lb .
4 In order to facilitate the analysis and design, the
midbodyregionorotherlocalpartsofashipcan
be treated as a thin box girder structure. Given
exact parameters of cracks, the simple formula
proposed in Eq. (3) can be used to predict the
residualultimatestrength
ofsimplifiedhullgirder
structuresundertorsionalloading.
198
ACKNOWLEDGMENTS
The present work is supported by the Chinese
Government Key Research Project KSHIP‐II Project
(Knowledge‐based Ship Design Hyper‐Integrated
Platform)No201335.
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